For requesting, clarifying, and comparing definitions of mathematical terms.

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103 views

question about definition Silverman's AEC

I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ ...
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280 views

why are curl and divergence defined the way they are?

Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and ...
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13 views

Quotient topology from Delta complex,

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_{\alpha} : \Delta^n \rightarrow X$ with n depending on index $\alpha$ such that 1)The restriction ...
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28 views

Definition of mathematical expression

According to wikipedia: "In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical ...
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40 views

Supply/demand digraph - understanding problem

I'm dealing with multiflows and I found in "Combinatorial Optimization - Part C" by Schrijver in Chapter 70 a good source. The definition of the multiflow problem involves so-called supply- and ...
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25 views

Set of 2-tuples with indices

I want to define the set of two tuples. Actually, I want to describe the degree distribution (or histogram) in terms of ordered tuples. The first item denotes the degree of a node. The second item ...
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32 views

Sup-norm of two functions

Suppose $\Omega$ is a bounded set and connected domain in $\mathbb{R}^n$. Consider the operator $L$ in $\Omega$ $L=a_{ij}(x)D_{ij}u+b_{i}(x)D_{i}u+c(x)u$ For $u \in C^{2}(\Omega)\cap ...
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29 views

Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic ...
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27 views

Is this definition correct regarding the logic of recursive sequences?

This is a self study. I come across with a second order sequence of reals defined by: $$φ_{m+2}=φ_{⌊f(φ_{m},φ_{m+1})⌋}..................(*)$$ where $⌊.⌋$ is the integer part function. Here $f$ is a ...
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19 views

Definition of derivative as best affine approximation

I'd like to try to redefine the derivative (in a way equivalent to the usual definition) of a function $f: U \subseteq R \to R$ to make it clear that the derivative $Df(a)$ is the linear part of the ...
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54 views

Definition: is a graph allowed to have a “dangling” edge without a vertex at its end(s)?

My textbook gives the following definition "a graph $G=(V,E)$ consisting of $V$, a nonempty set of vertices and $E$, a set of edges. Each edge has either one or two vertices associated with it." Now ...
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31 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...
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24 views

Different notions of Submanifold

There are three types of submanifolds discussed in my book. Let $M$ be a smooth manifold. Then 1.) An immersed submanifold of $M$ is a set $S\subseteq M$ such that $S=F(S)$, where $F:N\to M$ is an ...
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32 views

Why are these definitions equivalent for $\alpha=1$?

I am getting confused over what I can only assume is a technicality of a limit in the following definitions; Definition 1: Let $I=(a,b)\subset \mathbb{R}$ be an interval and $f, F:I\rightarrow ...
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10 views

What does “smoothly dependent” mean here?

Let $T\colon Y\times M\to Y$ be an operator, where $M$ is the parameter space and $Y$ is a normed space. $T$ is called smooth on $Y$ if both map $T$ and its derivative $T'(y)$ are ...
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39 views

What does maximal order mean in a group?

As written in Abstract Algebra by T. W. Judson: Lemma 13.4 : Let $G$ be a finite abelian $p-$group and suppose that $g ∈ G$ has maximal order. Then $G$ is isomorphic to $g × H$ for some subgroup ...
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61 views

Why not generalize the Intermediate Value Theorem

So I know the IVT says that given a continuous function on a closed interval [a,b] then if $f(a) < \gamma <f(b)$ then there is $c\in(a,b)$ such that $f(c) = \gamma$. Is there any reason for not ...
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65 views

Cluster Points of $S_{\Omega}$

Let $T: X \to X$ a continuous map on a compact set $X\subset \mathbb{R}$. A point $x \in X$ is non-wandering if for any open set $U \ni x$ there exists $n>0$ such that $T^{n}(U)\cap U \ne ...
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30 views

“Second kind” orthogonal polynomials and functions

Recently I've been doing reading in the subject of orthogonal polynomials on the real line (OPRL). Such OPs arise in solving the three-term recurrence relation $$x ...
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39 views

Is there anything called kernel space?

Here I am referring kernel as an integral operation.The wikipedia link is this https://en.wikipedia.org/wiki/Integral_transform My question is: consider the function insider the integral $f(t)$ is ...
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40 views

Notation/definition problem for commutative binary operation

I'm trying to describe/define the commutative binary operation on a three-element set which when the operands are the same, gives the same element and when they are different gives the element which ...
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45 views

Can we have different methods to estimate elements from Lp spaces?

Sorry if my question is vague. Consider I have some time samples and it is known to be summation of sinusoidal. Problem is to estimates the frequencies. Generally, Fast Fourier transform (FFT) is the ...
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14 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . On the other hand interpolation space which is defined in the wikipedia link: ...
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23 views

How is the line integration defined in the most general setting?

A natural generalization of Riemann-Stieltjes integration is Lebesgue integration. Some would say that the use of Lebesgue integration would be overkill when treating differentiable or continuous ...
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43 views

Open balls in the definition of a Euclidean submanifold

Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there ...
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19 views

Can we have extension of Mercer theorem to interpolation?

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
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73 views

Definition: vector or point belonging to an area

This is an applied problem, which I try to define mathematically. I have two vehicles, vehicle 1 is defined by the area, dependent on length $L$ and width $W$ of the vehicle, according to: ...
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16 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...
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59 views

Operator form $L^2$ space to$L^1$

Can we have an operator such that it transforms an element of $L^2$ to $L^1$? Is this a valid question or this is incorrect? We can consider the measure space as finite.
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118 views

Is there a special name for functors from a category C to a subcategory of C?

Is there a special name for functors from a category $C$ to a subcategory $S$ of $C$ where $S \ne C$? I'm using $\ne$ pretty informally above. I'd be happy with anything that reasonably captures the ...
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31 views

Any relation between Kernel methods and Variational methods?

I am familiar with kernel method, which is defined in the link: https://en.wikipedia.org/wiki/Kernel_method On the other hand I am familiar with variational methods which is defined in the link: ...
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14 views

definition of an affine F-variety where F is not algebraically closed

In the first chapter of his book "Linear Algebraic Groups", Springer considers a situation where $k$ is an algebraically closed field and $F$ is a subfield, and seeks to define the notion of an ...
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36 views

Help in understanding Bochner's theorem and Pontryagin duality theorem

I am trying to understand Bochner's theorem through wikipedia link https://en.wikipedia.org/wiki/Bochner's_theorem This refers to dual spaces of locally compact abelian group and leads to ...
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32 views

What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since ...
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17 views

On random functions taking values in the space of continous functions.

I upload the image of the passage that is unclear to me (Theoretical statistics by Keener): I do not understand how $W_1, W_2, \dots$ take values in $C(K)$. For every different realization of the ...
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32 views

What is a separable closure? And why the standard definition of algebraic closure is defind as so?

Instead of defining it as a subfield of algebraic closure, what would be the natural way to define it? Here's what I figured out: Let $F$ be a field and $E$ be an extension field of $F$. ...
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70 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
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53 views

Correct understanding of col and diag operators

In a scientific paper I am currently working with, a definition of $Col$ and $Diag$ operator is introduced: We use the operator $Col_{k\in K}(x_k)$ which stacks up its vector (or matrix) ...
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30 views

Definition of a minimal set of automorphisms generating an orbit?

By definition, the orbit of a vertex v in a graph G is the set of all vertices f (v) such that f is an automorphism of G. I wonder whether there is a definition for a minimal set S of automorphisms ...
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28 views

Are little-o and “error term” the same thing?

I'm reading The Elements of Infinitesimal Calculus by David A. Santos, and I stumbled upon this: 45 Definition Let $m$ and $n$ be non-zero natural numbers with $m < n$. We say that $x^n$ ...
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78 views

Why is the conditional probability treated as a definition in Kolmogorov's probability theory?

The conditional probability is defined as: $$P(A|B) = \frac {P(A \cap B)} {P(B)}$$ given that $$P(B) \neq 0$$ This is achieved based on our intuition, along with the Venn diagram description of the ...
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41 views

Definition - limit of a sequence - uses “rank”?

I have the following definition in my book, and was confused as to the context of the word "rank" here. The definition is as follows: A sequence $(u_n)_{n∈N}$ has limit $l ∈ R$ as $n → ∞$ (we also ...
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37 views

A quick question on the limsup of sets and logical connectives

I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we ...
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26 views

What is a “nonparametric function”?

I am looking for a formal definition of the term "nonparametric function." I understand the term and I use nonparametric regressions http://en.wikipedia.org/wiki/Nonparametric_regression but I ...
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50 views

What does rigoruous but non-technical mean?

Hi I find the above expression a bit confusing. I am considering buying a book and it says that it's a rigorous but non-technical introduction to optimal stochastic control. Could someone explain ? ...
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52 views

Proper usage for term, addend, factor, multiplicand, expression, formula

The definitions and usage of the following words seem to vary, depending on the source text: term addend factor multiplicand expression formula The words are being used in the context of ...
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66 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
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45 views

$k$-cube definition clarrification

If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube. What will $\partial c$ be? Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else? Definition - A singular $k$-cube on $U$ ...
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20 views

What is the $\bar{d}$-metric for translation-invariant measures?

I've often heard of the so called $\bar{d}$-metric for translation-invariant measures. I found something like $$ \bar{d}(m_1,m_2)=\inf\text{Prob}^m\left\{\eta(0)\neq\delta(0)\right\}, $$ where $m_1$ ...
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66 views

What is a coupling argument?

In an article I've read in a proof that distinguishes two cases something like: "the second case can be shown by an easy coupling argument using the first case." What is a coupling argument? Edit ...